Input Parameter Constraints

Examination of Figure 4.8 shows how, when coupled with the extrinsic parameters of distance and reddening that are applied as additive offsets to magnitude and color, certain combinations of input parameters produce redundant fits to the data. Therefore, where possible, independent measurements of the physical properties these parameters represent must constrain the inputs.

Metallicity

The faintness of the brightest RGB stars in both clusters limit spectroscopic studies to measuring the relative strength of the H + K calcium II lines at 3968.5Å and 3933.7Å respectively, the same technique and metallicity scale employed by Zinn & West (1984, ZW) hence the subscript “ZW”. This was calibrated, however, using high-dispersion, high SNR spectroscopic analysis of many globular clusters. Another commonly used metallicity scale is that of Carretta & Gratton (1997) distinguished with a “CG” subscript. They used different high-dispersion spectra than ZW to calibrate their results. Differences of∼0.2dex were seen between CG and ZW at intermediate metallicities but not at the highest or lowest [Fe/H]. Marín-Franch et al. (2009) lists all all MW globular clusters in both metallicity scales and provides transformation equations between the two. Suntzeff, Olszewski, & Stetson (1985) measured[Fe/H]ZW =−1.7±0.2for AM 1 based on the brightest two RGB stars, and Palma,

Kunkel, & Majewski (2000) obtain[Fe/H]ZW =−1.4±0.1for Pyxis based on multiple spectra

of the brightest star in the cluster.

Both spectroscopic measurements tend to the metal-rich end of the uncertainty range compared to photometric techniques. For instance, Sarajedini & Geisler (1996) find Pyxis to have [Fe/H] = −1.20±0.15using the simultaneous reddening and metallicity technique (SRM. Sarajedini 1994), and the best fitting isochrones for AM 1 by D08b use[Fe/H] =−1.5. Recall, however, the ridge line overlay technique discussed in §4.1. As shown in Figure 4.2, due to the steeper slope of the RGB, both AM 1 and Pyxis are more metal rich than M3, and the departure of the RGB of M3 at brighter magnitudes hints at a greaterαenrichment or a higher [Fe/H] in both target clusters. Though this is not a precise measurement, the close match with the slope and curvature of M5 indicates both target clusters are of similar metallicity and[α/Fe]. Using the ZW84 scale to be consistent with the spectroscopic measurements mentioned above, M3 and M5 have[Fe/H]ZW =−1.66and−1.38respectively

α Enhancement

Undertaking a comprehensive review of the measurements available at the time, Carney (1996, Table 3 in particular) showed that halo GCs share [α/Fe] at or very near +0.3. The most recent spectroscopic studies of the extreme outer halo, particularly the “typical” halo clusters Pal 3, Pal 4 and Pal 15, support Carney’s characterization (see Table 1.1). Unfortunately, the DSED does not include inputs at exactly this value, instead bracketing it by +0.1 on either side. The effect of [α/Fe] on the isochrone is to increase the slope and curvature of the RGB, particularly steepening the upper RGB and leaving the lower portion near the intersection of the SGB unaffected because of the H− effect noted earlier.

The choice of [α/Fe] = +0.4 for both clusters, while providing the best looking fit to the data, relies on the brightest stars along the RGB track of the isochrone to be actual members of the cluster. A kinematic study of the brightest two giants in AM 1 by Suntzeff et al. (1985, see Table III) shows that these stars’ radial velocities are consistent with each other as well as the radial velocity of the cluster as a whole providing some support to their inclusion as members of the cluster. Similarly, Palma et al. (2000, see Table 1) shows uniformity among the radial velocities of the brightest six stars in Pyxis. The results from both studies are listed in Table 4.4. The former study’s inclusion of only two stars, shown as red plus signs in the left panel of Figure 4.9, as well as greater spread in radial velocity than the latter, lends some caution to the brightest star being a member of AM 1. The magnitudes and colors for Pyxis in Table 4.4 ultimately come from Sarajedini & Geisler (1996), but they observed in the BR filter system preventing a direct identification of these stars in my CMD that uses the BV system. However, Palma et al. (2000) encounters the same problem and invokes color-color transformations to listV ≈17.77for the brightest star in their survey which they label Pyxis A. Lacking a (B−V) color, I chose the color for the star closest in magnitude among my data for the position of the red plus in the right panel of Figure 4.9. Palma et al. (2000) does not supply the V transformation for the remainder of the stars in his survey nor

Table 4.4. Radial Velocities of the Brightest Red Giants

AM 1 (Suntzeff et al. 1985) Pyxis (Palma et al. 2000) Star IDa _{V (B−V) v}

helio(km/s) Star IDa R (B−R) vhelio(km/s)

45 18.21 1.46 102 A 17.08 2.01 32.7 36 19.38 0.97 130 B 17.75 1.80 38.4 C 18.28 1.78 36.6 D 18.33 1.75 26.1 E 18.08 1.78 37.9 F 18.09 1.71 33.9

Note. — The uncertainties associated with the radial velocities for AM 1 and Pyxis are±9and ±4.6km/s respectively.

a_{Star identification assigned by the respective authors.}

cite the source of the color-color transformations, but they are all fainter in R than Pyxis A. Thus I lack strong support to conclude that the brightest two stars along the isochrone track are cluster members, but their membership is, at least, plausible.

Distance

Distance modulus relates the difference between the apparent, m, absolute magnitude,

M, of stars to their distance,D, in parsecs via

DMHB= (m−M)_HB= (VHB−MV(RR)) = 5 logD−5. (4.10)

The mean level of the horizontal branch, determined in §4.1 (see Figure 4.2), serves as the apparent magnitude, and the absolute magnitude of the HB is based on the compilation by Cacciari & Clementini (2003, CC03, Equation 6.12₎

MV(RR) = (0.23±0.04)[Fe/H] + (0.93±0.05) (4.11)

2_{This equation is actually a reprint from Chaboyer (1999, Equation 6) who cites the uncertainty on the offset}

term as±0.12. By applying a weighted mean of all the available techniques, Cacciari & Clementini reduce the uncertainty to the value listed in Equation 4.11.

where MV(RR) refers to the mean magnitude of the RR Lyrae variable stars in the cluster,

and is interchangeable with the mean magnitude of the HB.

Given that metallicity is a free parameter in the isochrone models as well as dependent variable on the absolute magnitude of the HB, finding a match between the isochrone distance modulus, denoted as DMiso, and the measured distance modulus correction, DMHB, lends

credence to the quality of the fit. As indicated in Figure 4.9, the isochrone and measured values of distance match nearly exactly. Looking back at Figure 4.2, the HB for Pyxis is poorly defined, and the uncertainty for VHB reflects the standard deviation of the mean

magnitude. The uncertainty for DMHB includes this as well as the uncertainties on the

coefficients in Equation 4.11 added by quadrature with the uncertainty on the mean RR Lyrae magnitude, MV(RR). The latter uncertainty requires some additional explanation.

Adopting the notation of Chaboyer (1999, Equation 1), we rewrite Equation 4.11 as

MV(RR) =α[Fe/H] +β (4.12)

where the coefficients have associated uncertainties σα and σβ. Employing the regular rules

of error propagation, we derive a total uncertainty for the mean RR Lyrae magnitude as

σ2_M V(RR) = ([Fe/H]σα) 2 + α σ[Fe/H] 2 +σ_β2. (4.13)

Using the metallicity measurement of Palma et al. (2000) of [Fe/H] = −1.4±0.1 as well as the statistical uncertainty on the mean horizontal branch magnitude shown in Figure 4.2, we find the total uncertainty σDMHB =±0.08.